Question

Standing by a pool filled with water, under what condition will you see a reflection of the scenery on the opposite side through total internal reflection of the light from the scenery? a) Your eyes are level with the water. b) You observe the pool at an angle of \(41.8^{\circ}\). c) There is no condition under which this can occur. d) You observe the pool at an angle of \(48.2^{\circ}\).

Short answer

a) Your eyes are level with the water. b) You observe the pool at an angle of 41.8 degrees. c) There is no condition under which this can occur. d) You observe the pool at an angle of 48.2 degrees. Answer: None of the given conditions will result in total internal reflection, as the observer's angle must be greater than the critical angle of 48.6 degrees for total internal reflection to occur.

Step-by-step solution

Determine the critical angle

First, we need to find the critical angle, which can be calculated using Snell's law. The refractive index of air is approximately \(1\) and for water, it is approximately \(1.33\). Therefore, using Snell's law, we can determine the critical angle: $$ \sin{\theta_c} = \frac{n_2}{n_1} = \frac{1}{1.33} $$ Now, we can calculate the critical angle: $$ \theta_c = \arcsin{\frac{1}{1.33}} \approx 48.6^{\circ} $$ So, the critical angle for total internal reflection is \(48.6^{\circ}\).

Analyze the given conditions

Now, let's analyze each of the given conditions: a) Your eyes are level with the water. In this case, the angle at which you observe the pool is \(0^{\circ}\), which is lesser than the critical angle of \(48.6^{\circ}\). Therefore, total internal reflection will not occur. b) You observe the pool at an angle of \(41.8^{\circ}\). In this case, the angle at which you observe the pool is lesser than the critical angle of \(48.6^{\circ}\). Therefore, total internal reflection will not occur. c) There is no condition under which this can occur. Based on the critical angle that we calculated, there could be conditions in which total internal reflection occurs. So, this option is incorrect. d) You observe the pool at an angle of \(48.2^{\circ}\). In this case, the angle at which you observe the pool is slightly smaller than the critical angle, \(48.2^{\circ} < 48.6^{\circ}\). Therefore, total internal reflection will not occur. As we can see, none of the given conditions satisfy the requirement for total internal reflection to occur. However, if any angle greater than the critical angle \(48.6^{\circ}\) were provided, then total internal reflection would occur in that case.

Key concepts

Critical Angle

Imagine you're gazing upon a tranquil pool of water, noticing the beautiful scenery on the opposite side. Sometimes, the surface of the water can appear like a perfect mirror, showcasing the reflections of trees and clouds. This phenomenon, called total internal reflection, occurs only when a very specific and crucial angle is surpassed. Here lies the importance of the 'critical angle': it marks the minimum angle of incidence at which light rays, attempting to move from a denser medium (like water) to a rarer one (like air), get reflected entirely back into the denser medium instead of refracting out.

Reaching or exceeding this critical angle means that no light escapes the water; it's all reflected internally. The critical angle is unique for each pair of media because it depends on their refractive indices. And as in our pool scenario, if you're not observing at an angle equal to or greater than the critical angle, you won't witness the total internal reflection of those serene images floating on the opposite side.

Snell's Law

Snell's law is like the rulebook for light rays travelling between different mediums. Put simply, Snell's law describes how light bends, or refracts, when it crosses the boundary between media with different refractive indices. Think of it as a ray of sunlight entering a pool of water: the light changes direction slightly, which is why things under water can look a bit offset from where they actually are.

The law can be written as \[\[\begin{align*} n_1 \times \text{sin}(\theta_1) = n_2 \times \text{sin}(\theta_2)d{align}where \(n_1\) and \(n_2\) are the refractive indices of the first and the second medium, and \(\theta_1\) and \(\theta_2\) are the angles of incidence and refraction, respectively. By manipulating Snell's law, we can calculate the critical angle for any given pair of media, which determines whether we'll observe refraction or the captivating illusions created by total internal reflection.

Refractive Index

The refractive index is fundamental when you're trying to predict how light will behave as it moves from air into water, or from any one material to another. It's a number without units that tells you how much light will bend. In essence, the refractive index gives you a measure of how much a material can slow down light compared to its speed in a vacuum (the speediest it can ever go!).
For example, water has a refractive index of around 1.33. This means that light travels about 1.33 times slower in water than in a vacuum. The bigger this number, the more the light slows down and bends when it strikes the material. This change in speed is what causes things to look out of place when submerged in water, and it's also the deciding factor for the critical angle we previously discussed.

Optics

The Science of Sight

Optics is the branch of physics that's all about light: how it moves, interacts with materials, and helps us perceive the world. It's not just about lenses and spectacles – it's the reason we can see at all! The study of optics covers a range of phenomena, from rainbows arching across the sky after a storm to the fiber optic cables transmitting data across continents at the speed of light. Understanding principles like the critical angle and Snell's law is crucial for applications in medicine, communications, and even in designing safer vehicles. Essentially, optics is a window into how we interact with light and, by extension, our entire visual experience.